How to Read a Logarithmic Scale
Determine whether you are reading a semi-log or log-log graph., Read the scale of the main divisions., Notice that the minor intervals are not evenly spaced.
Step-by-Step Guide
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Step 1: Determine whether you are reading a semi-log or log-log graph.
Graphs that represent rapidly growing data can use one-log scales or two-log scales.
The difference is in whether both the x-axis and y-axis use logarithmic scales, or only one.The choice depends on the amount of detail that you wish to display with your graph.
If numbers on one axis or the other grow or decrease exponentially, you may wish to use a logarithmic scale for that axis.
A logarithmic (or just “log”) scale has unevenly spaced grid lines.
A standard scale has evenly spaced grid lines.
Some data needs to be graphed on standard paper only, some on semi-log graphs, and some on log-log graphs.
For example, the graph of y=x{\displaystyle y={\sqrt {x}}} (or any similar function with a radical term) can be graphed on a purely standard graph, a semi-log graph, or log-log graph.
On a standard graph, the function appears as a sideways parabola, but the detail for very small numbers is difficult to see.
On the log-log graph, the same function appears as a straight line, and the values are more spread out for better detail.If both variables in a study include great ranges of data, you would probably use a log-log graph.
Studies of evolutionary effects, for example, may be measured in thousands or millions of years and might choose a logarithmic scale for the x-axis.
Depending on the item being measured, a log-log scale may be necessary. -
Step 2: Read the scale of the main divisions.
On a logarithmic scale graph, the evenly spaced marks represent the powers of whatever base you are working with.
The standard logarithms use either base 10 or the natural logarithm which uses the base e{\displaystyle e}. e{\displaystyle e} is a mathematical constant that is useful in working with compound interest and other advanced calculations.
It is approximately equal to
2.718.This article will focus on the base-10 logarithms, but the reading the natural logarithm scale operates in the same way.
Standard logarithms use base
10.
Instead of counting 1, 2, 3, 4… or 10, 20, 30, 40… or some other evenly spaced scale, a logarithm scale counts by powers of
10.
The main axis points are, therefore, 101,102,103,104{\displaystyle 10^{1},10^{2},10^{3},10^{4}} and so on.Each of the main divisions, usually noted on log paper with a darker line, is called a “cycle.” When specifically using based 10, you can use the term “decade” because it refers to a new power of
10. , If you are using printed logarithmic graph paper, you will notice that the intervals between the main units are not evenly spaced.
That is, for example, the mark for 20 would actually be placed about 1/3 of the way between 10 and
100.The minor interval marks are based on the logarithm of each number.
Therefore, if 10 is represented as the first major mark on the scale, and 100 is the second, the other numbers fall in between as follows: log(10)=1{\displaystyle log(10)=1} log(20)=1.3{\displaystyle log(20)=1.3} log(30)=1.48{\displaystyle log(30)=1.48} log(40)=1.60{\displaystyle log(40)=1.60} log(50)=1.70{\displaystyle log(50)=1.70} log(60)=1.78{\displaystyle log(60)=1.78} log(70)=1.85{\displaystyle log(70)=1.85} log(80)=1.90{\displaystyle log(80)=1.90} log(90)=1.95{\displaystyle log(90)=1.95} log(100)=2.00{\displaystyle log(100)=2.00} At higher powers of 10, the minor intervals are spaced in the same ratios.
Thus, the spacing between 10, 20, 30… looks like the spacing between 100, 200, 300… or 1000, 2000, 3000…. -
Step 3: Notice that the minor intervals are not evenly spaced.
Detailed Guide
Graphs that represent rapidly growing data can use one-log scales or two-log scales.
The difference is in whether both the x-axis and y-axis use logarithmic scales, or only one.The choice depends on the amount of detail that you wish to display with your graph.
If numbers on one axis or the other grow or decrease exponentially, you may wish to use a logarithmic scale for that axis.
A logarithmic (or just “log”) scale has unevenly spaced grid lines.
A standard scale has evenly spaced grid lines.
Some data needs to be graphed on standard paper only, some on semi-log graphs, and some on log-log graphs.
For example, the graph of y=x{\displaystyle y={\sqrt {x}}} (or any similar function with a radical term) can be graphed on a purely standard graph, a semi-log graph, or log-log graph.
On a standard graph, the function appears as a sideways parabola, but the detail for very small numbers is difficult to see.
On the log-log graph, the same function appears as a straight line, and the values are more spread out for better detail.If both variables in a study include great ranges of data, you would probably use a log-log graph.
Studies of evolutionary effects, for example, may be measured in thousands or millions of years and might choose a logarithmic scale for the x-axis.
Depending on the item being measured, a log-log scale may be necessary.
On a logarithmic scale graph, the evenly spaced marks represent the powers of whatever base you are working with.
The standard logarithms use either base 10 or the natural logarithm which uses the base e{\displaystyle e}. e{\displaystyle e} is a mathematical constant that is useful in working with compound interest and other advanced calculations.
It is approximately equal to
2.718.This article will focus on the base-10 logarithms, but the reading the natural logarithm scale operates in the same way.
Standard logarithms use base
10.
Instead of counting 1, 2, 3, 4… or 10, 20, 30, 40… or some other evenly spaced scale, a logarithm scale counts by powers of
10.
The main axis points are, therefore, 101,102,103,104{\displaystyle 10^{1},10^{2},10^{3},10^{4}} and so on.Each of the main divisions, usually noted on log paper with a darker line, is called a “cycle.” When specifically using based 10, you can use the term “decade” because it refers to a new power of
10. , If you are using printed logarithmic graph paper, you will notice that the intervals between the main units are not evenly spaced.
That is, for example, the mark for 20 would actually be placed about 1/3 of the way between 10 and
100.The minor interval marks are based on the logarithm of each number.
Therefore, if 10 is represented as the first major mark on the scale, and 100 is the second, the other numbers fall in between as follows: log(10)=1{\displaystyle log(10)=1} log(20)=1.3{\displaystyle log(20)=1.3} log(30)=1.48{\displaystyle log(30)=1.48} log(40)=1.60{\displaystyle log(40)=1.60} log(50)=1.70{\displaystyle log(50)=1.70} log(60)=1.78{\displaystyle log(60)=1.78} log(70)=1.85{\displaystyle log(70)=1.85} log(80)=1.90{\displaystyle log(80)=1.90} log(90)=1.95{\displaystyle log(90)=1.95} log(100)=2.00{\displaystyle log(100)=2.00} At higher powers of 10, the minor intervals are spaced in the same ratios.
Thus, the spacing between 10, 20, 30… looks like the spacing between 100, 200, 300… or 1000, 2000, 3000….
About the Author
Sara Wells
Writer and educator with a focus on practical pet care knowledge.
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